Difference between revisions of "One-parameter subgroup"
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− | + | ''of a Lie group | |
+ | over a normed field K '' | ||
− | + | An analytic homomorphism of the additive group of the field K | |
+ | into G , | ||
+ | that is, an analytic mapping $ \alpha : K \rightarrow G $ | ||
+ | such that | ||
− | + | $$ | |
+ | \alpha ( s + t) = \alpha ( s) \alpha ( t),\ s, t \in K. | ||
+ | $$ | ||
− | If | + | The image of this homomorphism, which is a subgroup of G , |
+ | is also called a one-parameter subgroup. If $ K = \mathbf R $, | ||
+ | then the continuity of the homomorphism \alpha : K \rightarrow G | ||
+ | implies that it is analytic. If K = \mathbf R | ||
+ | or \mathbf C , | ||
+ | then for any tangent vector X \in T _ {e} G | ||
+ | to G | ||
+ | at the point e | ||
+ | there exists a unique one-parameter subgroup \alpha : K \rightarrow G | ||
+ | having X | ||
+ | as its tangent vector at the point $ t = 0 $. | ||
+ | Here $ \alpha ( t) = \mathop{\rm exp} tX $, | ||
+ | t \in K , | ||
+ | where $ \mathop{\rm exp} : T _ {e} G \rightarrow G $ | ||
+ | is the [[Exponential mapping|exponential mapping]]. In particular, any one-parameter subgroup of the [[General linear group|general linear group]] G = \mathop{\rm GL} ( n, K) | ||
+ | has the form | ||
+ | |||
+ | $$ | ||
+ | \alpha ( t) = \mathop{\rm exp} tX = \ | ||
+ | \sum _ {n = 0 } ^ \infty | ||
+ | { | ||
+ | \frac{1}{n! } | ||
+ | } | ||
+ | t ^ {n} X ^ {n} . | ||
+ | $$ | ||
+ | |||
+ | If G | ||
+ | is a real Lie group endowed with a two-sidedly invariant pseudo-Riemannian metric or affine connection, then the one-parameter subgroups of G | ||
+ | are the geodesics passing through the identity e . | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1972) pp. Chapt. 2; 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G. Hochschild, "Structure of Lie groups" , Holden-Day (1965)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1972) pp. Chapt. 2; 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G. Hochschild, "Structure of Lie groups" , Holden-Day (1965)</TD></TR></table> |
Latest revision as of 08:04, 6 June 2020
of a Lie group G
over a normed field K
An analytic homomorphism of the additive group of the field K into G , that is, an analytic mapping \alpha : K \rightarrow G such that
\alpha ( s + t) = \alpha ( s) \alpha ( t),\ s, t \in K.
The image of this homomorphism, which is a subgroup of G , is also called a one-parameter subgroup. If K = \mathbf R , then the continuity of the homomorphism \alpha : K \rightarrow G implies that it is analytic. If K = \mathbf R or \mathbf C , then for any tangent vector X \in T _ {e} G to G at the point e there exists a unique one-parameter subgroup \alpha : K \rightarrow G having X as its tangent vector at the point t = 0 . Here \alpha ( t) = \mathop{\rm exp} tX , t \in K , where \mathop{\rm exp} : T _ {e} G \rightarrow G is the exponential mapping. In particular, any one-parameter subgroup of the general linear group G = \mathop{\rm GL} ( n, K) has the form
\alpha ( t) = \mathop{\rm exp} tX = \ \sum _ {n = 0 } ^ \infty { \frac{1}{n! } } t ^ {n} X ^ {n} .
If G is a real Lie group endowed with a two-sidedly invariant pseudo-Riemannian metric or affine connection, then the one-parameter subgroups of G are the geodesics passing through the identity e .
References
[1] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) |
[2] | J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) |
[3] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) |
Comments
References
[a1] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) |
[a2] | N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1972) pp. Chapt. 2; 3 |
[a3] | G. Hochschild, "Structure of Lie groups" , Holden-Day (1965) |
One-parameter subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_subgroup&oldid=11235